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Theorem rexlimdv3a 2526
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2523. (Contributed by NM, 7-Jun-2015.)
Hypothesis
Ref Expression
rexlimdv3a.1 ((𝜑𝑥𝐴𝜓) → 𝜒)
Assertion
Ref Expression
rexlimdv3a (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rexlimdv3a
StepHypRef Expression
1 rexlimdv3a.1 . . 3 ((𝜑𝑥𝐴𝜓) → 𝜒)
213exp 1163 . 2 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
32rexlimdv 2523 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 945  wcel 1463  wrex 2392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-3an 947  df-nf 1420  df-ral 2396  df-rex 2397
This theorem is referenced by:  resqrtcl  10741
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